Question

State and prove perpendicular axis theorem.
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Answer 1

Answer:

According to the theorem the moment of inertia of a plane lamina about any axis (OZ) perpendicular to plane is equal to sum of moment of inertia about any two  perpendicular axis (OX & OY) in the plane.

⇒ I = Iₓ + Iᵧ        [Formula]

Explanation:

Proof:

Suppose the lamina consist of n particles of mass m₁ + m₂ +.........+ mₙ and respective distance are r₁ + r₂ +........+ rₙ.

Moment of inertia about 'x' axis:

⇒ Iₓ = m₁y₁² + m₂y₂² +...........+mₙyₙ²    ______(1)

Moment of inertia about 'y' axis.

⇒ Iᵧ = m₁x₁² + m₂x₂² +...........+ mₙxₙ²   ______(2)

Moment of inertia about 'z' axis.

⇒ I = m₁r₁² + m₂r₂² +............+ mₙrₙ²   ______(3)

In general,

⇒ Iₓ = My²       _____(4)

⇒ Iᵧ = Mx²       _____(5)

Now, add equation (4) & (5), we get

⇒ Iₓ + Iᵧ = M(x² + y²)

⇒ Iₓ + Iᵧ = Mr²

⇒ Iₓ + Iᵧ = Iz

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Answer 2

$\textbf{Answer is in Attachment !}$